Private Counting from Anonymous Messages: Near-Optimal Accuracy with Vanishing Communication Overhead

2. The computer-implemented method of claim 1, wherein a combination of the first infinitely divisible distribution and the second infinitely divisible distribution produces a combined distribution that has both positive and negative support.

3. The computer-implemented method of claim 2, wherein the combination of the first infinitely divisible distribution and the second infinitely divisible distribution comprises a difference between the first infinitely divisible distribution and the second infinitely divisible distribution.

4. The computer-implemented method of claim 2, wherein the combined distribution comprises a Discrete Laplace distribution.

5. The computer-implemented method of claim 1, wherein the first infinitely divisible distribution is equal to the second infinitely divisible distribution.

6. The computer-implemented method of claim 1, wherein one or more of the first infinitely divisible distribution, the second infinitely divisible distribution, and the third infinitely divisible distribution respectively comprise respective negative binomial distributions.

7. The computer-implemented method of claim 1, wherein one or more of the first infinitely divisible distribution, the second infinitely divisible distribution, and the third infinitely divisible distribution respectively comprise Poisson distributions.

8. The computer-implemented method of claim 1, further comprising repeating the method of any preceding claim for at least a subset of an ordered plurality of private values, wherein a respective index value for each private value in the subset of the ordered plurality of private values is concatenated onto each message of the multiset of messages generated for such private value.

9. The computer-implemented method of claim 8, wherein the method further comprises randomly selecting the number of private values included in the subset of the ordered plurality of private values from a fourth sampling distribution.

10. The computer-implemented method of claim 8, wherein repeating the method of any preceding claim for at least the subset of an ordered plurality of private values comprises repeating, computationally in parallel, the method of any preceding claim for each private value in at least the subset of an ordered plurality of private values.

11. The computer-implemented method of claim 1, wherein the private value is a binary value.

13. The computer-implemented method of claim 12, wherein the infinitely divisible distribution comprises a Poisson distribution.

14. The computer-implemented method of claim 12, wherein the infinitely divisible distribution comprises a negative binomial distribution.

16. The computer-implemented method of claim 15, wherein the first sampling distribution subtracted from itself comprises a Discrete Laplace distribution.

17. The computer-implemented method of claim 15, wherein the second sampling distribution comprises a negative binomial distribution.

18. The computer-implemented method of claim 15, wherein the first number of increment messages and the second number of decrement messages have a sum equal to Discrete Laplacian noise.

19. The computer-implemented method of claim 15, wherein the private value comprises an integer value in a range from zero to a maximum user value.

20. The computer-implemented method of claim 19, wherein the zero sum noise messages are selected from a multiset such that values of the multiset are bounded by the maximum user value and such that the sum of all values in the multiset is equal to zero.